In video signal expansion and compression systems it is desired to perform a time scaling of discrete-time signals. However, in order to realize a general scaling of the discrete-time axis for the transformation y(m)=x(.alpha..multidot.m), a major problem arises; the input signal x(.alpha..multidot.m) is undefined for non-integral values of its index (.alpha..multidot.m). Interpolation is the solution to this problem. When the output requires a sample of the input at a time index with an undefined input value, an interpolated value must be supplied. A large body of literature exists on the topic of interpolation, but several points are worth noting: (a) interpolation which uses a large number of input samples weighted with a sin (x)/x function provides accurate results but is costly to implement for consumer apparatus; (b) sample-and-hold interpolation is the easiest to implement but has generally poor performance; (c) linear interpolation is relatively easy to implement and provides superior performance to the sample-and-hold technique, but exhibits increasing attenuation with increasing signal frequency; and (d) higher order interpolation provides superior performance to linear interpolation but exhibits nonlinearities.
T. J. Christopher in U.S. Pat. No. 4,694,414 described a relatively non-complex interpolator which exhibits relatively accurate performance and which is realized with a parallel combination of a two-point linear interpolator and a phase compensation filter. The compensation filter has a transfer function H.sub.(z) given by EQU H.sub.(z) =-1+z.sup.-1 +z.sup.-2 -z.sup.-3) (1)
where z is the conventional "z" transform variable and the exponents thereof correspond to the number of sample intervals. It should be noted that the weighting coefficients are either plus or minus one. The amplitude characteristic of the filter A(.phi.) is given by EQU A(.PHI.)=2 cos (.PHI./2)-2 cos (3.PHI./2) (2)
where .PHI. represents frequency in radians per second.
The compensation filter is cascaded with a gain element programmed with estimated gain values corresponding to the possible positions at which the system is designed to produce interpolated values. These gain values are derived by calculating the response error at a particular signal frequency. Since these gain values are not a function of frequency the response of the Christopher system contains residual errors.
It is an object of the present invention to provide a relatively simple interpolation circuit, which has a substantially accurate response characteristic for all signal frequencies up to the Nyquist sampling limit of the system.